1. 化學數學（一）
The Mathematics for Chemists (I) (Fall Term, 2004) (Fall Term, 2005) (Fall Term, 2006) Department of Chemistry National Sun Yat-sen University

2. Chapter 3 Vector Algebra and Analysis
Definition
Scalar (dot) product
Vector (cross) product
Scalar and vector fields
Applications

3. Content covered in the textbook: Chapter 16
Assignment:
pp :
15,17,18,24,25,30,32,35,40,41,
43,47,48,55,58,60

4. Definition (naïve) B a a=AB A
Vectors are a class of quantities that require both magnitude and direction for
their specification. a
Terminal point
Terminal point A
a=AB B
Initial point
Initial point
Unit vector: a vector of unit length.
Null vector: a vector of zero length. (its direction is meaningless.)

5. Examples of Vectors Position, velocity, angular velocity, acceleration
Force, torque, momentum, angular momentum
Electric and magnetic fields, electric and magnetic dipole moments,

6. Vector Algebra a b Equality: a b a+b Addition: a b a+b -b a-b
Subtraction:
Scalar multiplication: a
1.5a -a
-0.5a

7. Example Show that the diagonals of a parallelogram bisect each other.
We need to show that the midpoints of OD and AB coincide. O a b A B D C C’

8. Classroom Exercise
Show that the mean of the position vectors of the vertices of a triangle is the position
vector of the centroid of the triangle. O A B C a b X B O A C a b X C’
C’’

9. Components and Decompositionθ a O N P x y i j a
axi
ayj
x=(2,3),y=(4.2,-5.6)

10. Components and Decomposition (in 3D Space)
axi
azk
ayj

11. Vector Algebra Restated
Equality:
Addition:
Subtraction:
Scalar multiplication:

12. Example

13. The Center of Mass (Gravity)

14. Dipole Moments -q r1 r Dependence of reference frame: q r2
If the total charge Q is zero (e.g., in a molecule), then
Total dipole moment:

15. Electric dipole moments

16. Symmetry and Dipole Moment
The total dipole moment of a tetrahedron: r4
r1=(a,a,a), r2=(a,-a,-a),
r3=(-a,a,-a), r4=(-a,-a,a) r1 r2 r3

17. Base Vectors
Orthogonal basis:
axi
azk
ayj
Nonorthogonal basis:

18. Classroom Exercise

19. Scalar Differentiation of a VectorB O
a(t)
a(t+Δt) Δa

20. Parametric Representation of a Curvex y z
r(t) C a b t

21. Position, Velocity, Momentum, Acceleration Newton’s Second Law
Speed:
Linear momentum:
Kinetic energy:
Acceleration:
Newton’s second law:

22. Classroom Exercise
Write the expression of momentum in terms of the planar polar coordinates

23. The Scalar (Dot) Product
Proof: θ a b O B A

24. Example

25. θ a b O B A b b θ a θ O a O If
Classroom exercise

26. Orthogonal and Coincident

27. Cartesian Base Vectors
Orthogonality:
axi
azk
ayj
Normalization (unit length):

28. Force and Work

29. Force and Work: General CaseB O
r(t)
r+Δr Δr
F(r)
F(r+Δr)

30. Charges in an Electric Fieldθ μ E

31. Magnetic Moment in a Magnetic Fieldθ m B

32. HOW DO YOU KNOW THEY ARE PARALLEL WITH EACH OTHER?

33. The Vector (Cross) Product
A new vector can be constructed from two given vectors: a b θ
bsinθ
Its magnitude:
Its direction: v a b A B
C=AxB
Right-hand rule:

34. Important properties v -v Anti-commutative: b Nonassociative: a
(Proof to be given later)
Classroom exercise:
If the cross product of two vectors is a zero vector,
they must be parallel or antiparallel to each other.

35. In Cartesian Basis
axi
azk
ayj

36. Example

37. Classroom Exercise
Calculate the cross product of above two vectors using

38. Application: Moment of Force (Torque)θ d A

39. An Electric Dipole in an Electric Field-q q r1 r2 O μ E T

40. A Magnetic Dipole in a Magnetic Fieldr B
-qm qm r1 r2 O m B T

41. Angular Velocity
In a plane: v ω r O
General case: θ
rsinθ O r v ω

42. Exercise
Classroom exercise

43. Exercise

46. Angular Momentum ω r p θ r O A d A special case: ω is perpendicular r:
(moment of inertia)

47. Conservation of Angular Momentum
If T=0, angular momentum
is conserved. m B T
For nuclear spins:
NMR measures how fast a nuclear spin precesses.

48. Scalar and Vector Fields

49. The Gradient of a Scalar Field
Vector differential operator
The gradient of a scalar field is a vector.

50. The Meaning of the Gradient
f(r+dr)
f(r) dr
Gradient is a convenient vector expression of
the derivative of multi-variable functions.

51. Example: Gradient

52. Example: Gradient as Force
Force is the negative of the gradient of the potential energy.

53. Example: Gradient as Forceq1 q2 r q1 r E

54. The Divergence of a Vector Field
Laplacian operator
Laplace’s Equation

55. Examples: Divergence The divergence is a measure of flux density:
the amount of ‘something’ flowing out of a
unit volume per second.

56. The Curl of a Vector Field

58. Classroom Exercise

59. Physical Meaning of curl (rot)
The curl of a velocity field is angular velocity field (x2).

60. Example: Curl
The curl of the velocity filed is a measure of the circulation of fluid around the point.

61. Supplementary (Not Required)
You may win 5 points for filling in the details here!

62. Major Theorems in Integration
Fundamental: b a L S
Green:
Stokes: V S
Gauss:

63. Major Theorems in Integrationb a L S V S
(Stokes’s theorem)
The general form: ∂M M

64. Classroom Exercise?
Find out the result of the following integration by using Gauss’s theorem: 1 x y z

65. The Maxwell Equations Electrostatics Electrodynamics
Related to the light speed c
In a medium:

66. Vector Spaces … Inner product space Inner (scalar) product:
Norm (length):

67. You as a vector in a high dimensional space
Name
Gender
Birth date
Birth place ID
Student ID
Height
Weight
Favorite drink
Favorite music …. …

68. Operations of Vectors in Fortran
Array
Loop: Dot/cross product
Gradient
Divergence
Curl
General vector spaces
Subroutine

69. Dot Product of Vectors
Write a program to calculate the dot product of any two vectors.
C Program for calculating dot product of any two vectors c
program dotpro1
parameter(n1=3)
real a1(n1),a2(n1)
write(6,*)'Input three components of first vector:'
read(5,*)(a1(i),i=1,n1)
write(6,*)'Input three components of second vector:'
read(5,*)(a2(i),i=1,n1)
dot=0.
DO 10,I = 1,n1
dot=dot+a1(i)*a2(i)
CONTINUE
print *,'The dot product is ', dot
print *,'Next calculation (0/1)?'
read(5,*)i
if(i.eq.1) goto 1
stop
end

70. Cross Product of Vectors
Write a program to calculate the cross product of any two vectors.
C Program for calculating cross product of any two vectors c
program crossp1
parameter(n1=3)
real a1(n1),a2(n1),crossp(n1)
write(6,*)'Input three components of first vector:'
read(5,*)(a1(i),i=1,n1)
write(6,*)'Input three components of second vector:'
read(5,*)(a2(i),i=1,n1)
CROSSP(1)=a1(2)*a2(3)-a1(3)*a2(2)
CROSSP(2)=a1(3)*a2(1)-a1(1)*a2(3)
CROSSP(3)=a1(1)*a2(2)-a1(2)*a2(1)
print *,'The cross product is \n', (crossp(i),i=1,3)
print *,'Next calculation (0/1)?'
read(5,*)i
if(i.eq.1) goto 1
stop
end

71. Cross Product of Vectors
Write a program to calculate the following vector operations.
C Program for calculating cross product of any three vectors
program crossp2
parameter(n1=3)
real a1(n1),a2(n1),a3(n1),crossp(n1)
real tmp(n1)
write(6,*)'Input three components of first vector:'
read(5,*)(a1(i),i=1,n1)
write(6,*)'Input three components of second vector:'
read(5,*)(a2(i),i=1,n1)
write(6,*)'Input three components of third vector:'
read(5,*)(a3(i),i=1,n1)
tmp1=0.0
tmp2=0.0
DO 10,I = 1,n1
tmp1=tmp1+a1(i)*a2(i)
tmp2=tmp2+a1(i)*a3(i)
CONTINUE
do 20 i=1,n1
CROSSP(i)=tmp2*a2(i)-tmp1*a3(i)
continue
print *,'The cross product is \n', (crossp(i),i=1,3)
print *,'Next calculation (0/1)?'
read(5,*)i
if(i.eq.1) goto 1
stop
end

72. Gradient of a Scalar Field
Write a program to calculate the gradient of a scalar function.

73. C Program for calculating the gradient of any scalar function
program grdt1
parameter(n1=3)
real grt(n1)
write(6,*) 'please input the position (x,y,z):'
read(5,*)x,y,z
dx=0.001
dy=0.001
dz=0.001
x2=x+dx
y2=y+dy
z2=z+dz
f1=x*sin(x+y)+2.0*y*z*exp(x)+3.0*z*z
f2=x2*sin(x2+y)+2.0*y*z*exp(x2)+3.0*z*z
fx=(f2-f1)/dx
f2=x*sin(x+y2)+2.0*y2*z*exp(x)+3.0*z*z
fy=(f2-f1)/dy
f2=x*sin(x+y)+2.0*y*z2*exp(x)+3.0*z2*z2
fz=(f2-f1)/dz
grt(1)=fx
grt(2)=fy
grt(3)=fz
print *,'The gradient at (', x,y,z, ') is \n', (grt(i),i=1,3)
print *,'Calculating the gradient of next point (0/1)?'
read(5,*)i
if(i.eq.1) goto 1
stop
end

74. The Divergence of a Vector Field
Write a program to calculate the divergence of a vector field.

75. C Program for calculating the divergence of any vector field
program divg1
parameter(n1=3)
real vecf(n1)
write(6,*) 'please input the position (x,y,z):'
read(5,*)x1,y1,z1
dx=0.0001
dy=0.0001
dz=0.0001
x2=x1+dx
y2=y1+dy
z2=z1+dz
vx1=x1*x1
vx2=x2*x2
divgx=(vx2-vx1)/dx
vy1=x1*cos(x1+x1*y1+x1*y1*z1)
vy2=x1*cos(x1+x1*y2+x1*y2*z2)
divgy=(vy2-vy1)/dy
vz1=2.*y1+6.*z1+exp(-y1*z1)
vz2=2.*y1+6.*z2+exp(-y1*z2)
divgz=(vz2-vz1)/dy
divg=divx+divy+divz
print *,'The divergence at (', x,y,z, ') is \n', divg
print *,'Calculating the gradient of next point (0/1)?'
read(5,*)i
if(i.eq.1) goto 1
stop
end

76. The Curl of a Vector Field
Write a program to calculate the rot of a vector field.

77. Using Subroutines (Go to Chapter 2)

78. Cross Product of Vectors
Write a program to calculate the following vector operations by
calling subroutine for calculating cross product of two vectors.

79. C Program for calculating cross product of any two vectors
program crossp1
parameter(n1=3)
real a1(n1),a2(n1),crossp(n1)
write(6,*)'Input three components of first vector:'
read(5,*)(a1(i),i=1,n1)
write(6,*)'Input three components of second vector:'
read(5,*)(a2(i),i=1,n1)
CROSSP(1)=a1(2)*a2(3)-a1(3)*a2(2)
CROSSP(2)=a1(3)*a2(1)-a1(1)*a2(3)
CROSSP(3)=a1(1)*a2(2)-a1(2)*a2(1)
print *,'The cross product is \n', (crossp(i),i=1,3)
print *,'Next calculation (0/1)?'
read(5,*)i
if(i.eq.1) goto 1
stop
end
C Subroutine for calculating cross
C product of any two vectors C
subroutine(a,b,c,n)
real a(n),b(n),c(n)
c(1)=a1(2)*a2(3)-a1(3)*a2(2)
c(2)=a1(3)*a2(1)-a1(1)*a2(3)
c(3)=a1(1)*a2(2)-a1(2)*a2(1)
return
end

80. C Program for calculating the cross
C product of any number of vectors
program crossp
parameter(n=3,n1=3)
real a1(n),a2(n),a3(n),TMP1(1),TMP2(N)
open(20,file=‘crossp.1’,err=9999)
read(20,*)(a1(i),i=1,n)
read(20,*)(a2(i),i=1,n)
read(20,*)(a3(i),i=1,n)
close(20)
call vcross(a2,a3,tmp1,n)
call vcross(a1,tmp1,tmp2,n)
open(30,file=‘crossp.2’,err=9999)
write(30,*)(tmp2(i),i=1,n)
CLOSE(30)
STOP
END
C Subroutine for calculating cross
C product of any two vectors C
subroutine(a,b,c,n)
real a(n),b(n),c(n)
c(1)=a1(2)*a2(3)-a1(3)*a2(2)
c(2)=a1(3)*a2(1)-a1(1)*a2(3)
c(3)=a1(1)*a2(2)-a1(2)*a2(1)
return
end

81. Cross Product of Vectors
Write a program to calculate the following vector operations by
calling subroutine for calculating cross product of two vectors.

82. C Subroutine for calculating cross C product of any two vectors C
C Program for calculating the cross
C product of any number of vectors
program crossp6
parameter(n=3)
real a1(n),a2(n),a3(n),a4(n),a5(n),a6(n),a7(n)
real TMP1(1),TMP2(N),TMP3(N),TMP4(N)
real TMP5(n),TMP6(n)
open(20,file=‘crossp6.1’,err=9999)
read(20,*)(a1(i),i=1,n)
read(20,*)(a2(i),i=1,n)
read(20,*)(a3(i),i=1,n)
read(20,*)(a4(i),i=1,n)
read(20,*)(a5(i),i=1,n)
read(20,*)(a6(i),i=1,n)
read(20,*)(a7(i),i=1,n)
close(20)
call vcross(a6,a7,tmp1,n)
call vcross(a5,a6,tmp2,n)
call vcross(a3,a4,tmp3,n)
call vcross(tmp1,tmp2,tmp4,n)
call vcross(tmp3,tmp4,tmp5,n)
call vcross(a1,tmp5,tmp6,n)
open(30,file=‘crossp6.2’,err=9999)
write(30,*)(tmp6(i),i=1,n)
CLOSE(30)
STOP
END
C Subroutine for calculating cross
C product of any two vectors C
subroutine(a,b,c,n)
real a(n),b(n),c(n)
c(1)=a1(2)*a2(3)-a1(3)*a2(2)
c(2)=a1(3)*a2(1)-a1(1)*a2(3)
c(3)=a1(1)*a2(2)-a1(2)*a2(1)
return
end

83. General Vector Spaces
Write a subroutine to calculate the inner product of two vectors in any dimension.

84. C Program for calculating dot product of any two vectors
program dotpro1
parameter(n1=3)
real a1(n1),a2(n1)
write(6,*)'Input three components of first vector:'
read(5,*)(a1(i),i=1,n1)
write(6,*)'Input three components of second vector:'
read(5,*)(a2(i),i=1,n1)
dot=0.
DO 10,I = 1,n1
dot=dot+a1(i)*a2(i)
CONTINUE
print *,'The dot product is ', dot
print *,'Next calculation (0/1)?'
read(5,*)i
if(i.eq.1) goto 1
stop
end
C subroutine for calculating dot product of any two vectors
subroutine vdot(a,b,n)
real a(n),b(n)
dot=0.
DO 10,I = 1,n1
dot=dot+a(i)*b(i)
CONTINUE
return
end

85. C program for calculating the inner product of two vectors
program innerp1
parameter(n=10)
do 10 i=1,n
a1(i)=i*1.2
a2(i)=-2.*+i*i
continue
call vdot(a1,a2,dot,n)
print *,'The inner product of two vectors is \n', dot
stop
end
C subroutine for calculating inner product of any two vectors
subroutine vdot(a,b,dot,n)
real a(n),b(n)
dot=0.
DO 10,I = 1,n1
dot=dot+a(i)*b(i)
CONTINUE
return